COVERING PROPERTIES OF IDEALS
MAREK BALCERZAK, BARNABÁS FARKAS, AND SZYMON GLA̧B
Abstract. M. Elekes proved that any infinite-fold cover of a σ-finite measure space
by a sequence of measurable sets has a subsequence with the same property such that
the set of indices of this subsequence has density zero. Applying this theorem he gave
a new proof for the random-indestructibility of the density zero ideal. He asked about
other variants of this theorem concerning I-almost everywhere infinite-fold covers of
Polish spaces where I is a σ-ideal on the space and the set of indices of the required
subsequence should be in a fixed ideal J on ω.
We introduce the notion of the J-covering property of a pair (A, I) where A is a σalgebra on a set X and I ⊆ P(X) is an ideal. We present some counterexamples, discuss
the category case and the Fubini product of the null ideal N and the meager ideal
M. We investigate connections between this property and forcing-indestructibility of
ideals. We show that the family of all Borel ideals J on ω such that M has the Jcovering property consists exactly of non weak Q-ideals. We also study the existence
of smallest elements, with respect to Katětov-Blass order, in the family of those ideals
J on ω such that N or M has the J-covering property. Furthermore, we prove a general
result about the cases when the covering property “strongly” fails.
1. Introduction
We will discuss the following result due to Elekes [8].
Theorem 1.1. Let (X, A, µ) be a σ-finite measure space and let (An )n∈ω be a sequence
of sets from A that covers µ-almost every x ∈ X infinitely many times. Then there
exists a set M ⊆ ω of asymptotic density zero such that (An )n∈M also covers µ-almost
every x ∈ X infinitely mamy times.
Applying this result, Elekes gave a nice new proof for the fact that the density zero
ideal is random-indestructible. He asked about other variants of this theorem, in particular, he asked whether the measure case could be replaced by the Baire category
case.
2010 Mathematics Subject Classification. 03E05, 03E15.
Key words and phrases. ideals, infinite-fold covers, covering properties, forcing-indestructibility, meager sets, null sets, Katětov-Blass order, Borel determinacy, Borel ideals, P-ideals, Fubini product.
The first author was supported by the Polish Ministry of Science and Higher Education Grant No
N N201 414939 (2010-2013); the second author was supported by Hungarian National Foundation for
Scientific Research grant nos. 68262, 83726 and 77476; the third author was supported by the Polish
Ministry of Science and Higher Education Grant No IP2011 014671 (2012-2014).
1
2
MAREK BALCERZAK, BARNABÁS FARKAS, AND SZYMON GLA̧B
For an ideal I ⊆ P(ω), we assume that ω ∈
/ I and Fin ⊆ I where Fin stands for the
ideal of finite subsets of ω. An ideal I is tall if each infinite subset of ω contains an
infinite element of I. Clearly, an ideal J on ω is tall iff its dual filter J∗ does not have
a pseudointersection, that is a set X ∈ [ω]ω such that X ⊆∗ Y for each Y ∈ J∗ where
X ⊆∗ Y means that X \ Y is finite.
An ideal J on ω is called a P-ideal whenever for every sequence of sets En ∈ I (n ∈ ω),
there is a set E ∈ I such that En ⊆∗ E for each n.
Each ideal on ω can be treated as a subset of the Cantor space 2ω via the standard
bijection between 2ω and P(ω), so we can talk about Borel, Fσ , analytic, meager. . .
ideals.
We will need the following very useful characterization of meager ideals:
Theorem 1.2. ([16], [6, Theorem 4.1.2]) An ideal J on ω is meager if, and only if there
is an partition (Pn )n∈ω of ω into finite sets (or even intervals) such that each element
of J contains only finitely many Pn ’s.
If I is an ideal on a set X then let I ∗ = {X \ A : A ∈ I} its dual filter and let
I + = P(X) \ I the set of I-positive subsets of X. If Y ⊆ X and Y ∈ I + , then the
restriction of I to Y is the following ideal on Y : I Y = {Y ∩ A : A ∈ I}.
More informations about ideals on ω can be found e.g. in [13].
Elekes discovered that these covering properties have an effect on forcing indestructibility of ideals. Assume J is a tall ideal on ω and P is a forcing notion. We say that J
is P-indestructible if P ∃ A ∈ J |Ẋ ∩ A| = ℵ0 for each P-name Ẋ for an infinite subset of
ω, i.e. in V P the ideal generated by J is tall. This property has been widely studied for
years. For general characterization theorems about forcing indestructibility of ideals see
[9] and [7]. In these papers the authors studied a very general class of forcing notions,
namely forcing notions of the form PI = Borel(X) \ I ordered by inclusion where X is
a Polish space and I is a σ-ideal on X (with Borel base). For instance, the Cohen and
the random forcing can be represented of this form by C = PM and B = PN where M is
the σ-ideal of meager subsets and N is the σ-ideal of null subsets (with respect to the
Lebesgue measure) of the real line (or 2ω or ω ω ).
We will need the following classical Borel ideals on ω:
The density zero ideal,
|A ∩ n|
=0
Z = A ⊆ ω : lim
n→∞
n
is a tall Fσδ P-ideal.
The summable ideal,
(
I1/n =
A ⊆ ω:
X
n∈A
1
<∞
n+1
)
COVERING PROPERTIES OF IDEALS
3
is a tall Fσ P-ideal and clearly I1/n ( Z.
Let ED be the eventually different ideal, that is
n
o
ED = A ⊆ ω × ω : lim sup |(A)n | < ∞
n→∞
where (A)n = {m ∈ ω : (n, m) ∈ A}. Let ∆ = {(n, m) ∈ ω × ω : m ≤ n} and
EDfin = ED ∆. These two ideals are tall Fσ non P-ideals.
At last, the Fubini-product of Fin by itself
Fin ⊗ Fin = A ⊆ ω × ω : ∀∞ n |(A)n | < ∞ .
This is a tall Fσδσ non P-ideal.
Furthermore, we will use the Katětov-Blass (KB) order on ideals: J0 ≤KB J1 iff there
is a finite-to-one function f : ω → ω such that f −1 [A] ∈ J1 for each A ∈ J0 .
In Section 2 we introduce a general covering property of a pair (A, I) with respect to
an ideal on ω where A is a σ-algebra and I is an ideal on its underlying set. This property
is a natural generalization of the interaction between the pair (measurable sets, ideal of
measure zero sets) and the density zero ideal proved by Elekes. We discuss this notion,
its connection to the star-uniformity of ideals on ω and to the Katětov-Blass order. We
give some negative results showing that in certain cases the respective a.e.-subcovers do
not exist on ω ω , and in general, we show that ideals with property (M) cannot have any
J-covering properties. We investigate covering properties of N ⊗ M. And at last, we
present the general effect of the covering property on forcing indestructibility of ideals.
In Section 3, applying a result of C. Laflamme on filter games, we characterize the
category case, namely we prove that if J is Borel, then M has the J-covering property
iff EDfin ≤KB J. In particular, we answer the question from [8] about the category case
of Elekes’ theorem. We also present some examples which show that our implications
are not reversible.
In Section 4 we discuss the existence of KB-smallest elements in the family of those
ideals J on ω such that N or M has the J-covering property. We show that Z is not
the KB-smallest in the measure case even among the Borel ideals. We prove that if
t = c, then there is no KB-smallest element of these families. At last, we investigate the
generic ideal on ω in the Cohen model.
In Section 5 we describe a class of ideals I on R (in general, on Polish groups) for
which the J-covering property fails in a strong fashion.
2. The J-covering property
We can consider the following abstract setting.
4
MAREK BALCERZAK, BARNABÁS FARKAS, AND SZYMON GLA̧B
Definition 2.1. Let X be an arbitrary set and I ⊆ P(X) be an ideal of subsets of X.
We say that a sequence (An )n∈ω of subsets of X is an I-a.e. infinite-fold cover of X if
x ∈ X : {n ∈ ω : x ∈ An } is finite ∈ I , i.e. lim sup An ∈ I ∗ .
n∈ω
Of course, the sequence (An ) above can be indexed by any countable infinite set. Assume
furthermore that A is a σ-algebra of subsets of X and J is an ideal on ω. We say that
the pair (A, I) has the J-covering property if for every I-a.e. infinite-fold cover (An )n∈ω
of X by sets from A, there is a set S ∈ J such that (An )n∈S is also an I-a.e. infinite-fold
cover of X.
If A is clear from the context (e.g. A = Borel(X) if X is a Polish space), then we
will write simply “I has the J-covering property” instead of “(A, I) has the J-covering
property”. Furthermore, we will use the following notation:
CP(I) = CP(A, I) = J : (A, I) has the J-covering property ,
and if Γ is a class (or property) of ideals then let CPΓ (I) = CP(I) ∩ Γ. For example,
we can talk about CPBorel (I), that is, the family of those Borel ideals J such that I has
the J-covering property.
First of all, we list some easy observations on this definition:
Observations 2.2.
(1) Clearly, in the previous definition it is enough to check infinite-fold covers instead of I-a.e. infinite-fold covers. Observe that, if I1 ⊆ I2 and (A, I1 ) possesses
the J-covering property, then (A, I2 ) also possesses it. In other words, I1 ⊆ I2
implies CP(I1 ) ⊆ CP(I2 ).
(2) In this context, Elekes’ theorem says that, if (X, A, µ) is a σ-finite (complete)
measure space, then (A, Nµ ) has the Z-covering property (i.e. Z ∈ CP(Nµ ))
where Nµ = {H ∈ A : µ(H) = 0}.
(3) It is trivial that, if a pair (A, I) has the J-covering property, then J must be
tall. In the study of covering properties of pairs (Borel(X), I) where I is a“nice”
σ-ideal on the Polish space X, interesting ideals on ω must have a bit stronger
property than tallness. If J is a tall ideal on ω then the star-uniformity of J is
the following cardinal:
non∗ (J) = min |H| : H ⊆ [ω]ω and @ A ∈ J ∀ H ∈ H |A ∩ H| = ω .
We claim that, if (A, I) has the J-covering property and A\I contains infinite
antichains (in the forcing sense), then non∗ (J) > ω.
S
Simply let (Bk )k∈ω be a partition of X = A into I-positive sets from A,
and assume on the contrary that (Yk )k∈ω is a sequence in [ω]ω such that ∀ S ∈ J
∃ k ∈ ω |S ∩ Yk | < ω. W.l.o.g. we can assume that this sequence is a partition
COVERING PROPERTIES OF IDEALS
5
of ω. Define An = Bk if n ∈ Yk . Then (An )n∈ω is infinite-fold cover, since any
Bk appears infinitely many times in this sequence. Take any S ∈ J. Then there
is k such that S ∩ Yk is finite, in particular Bk appears only finitely many times
in (An )n∈S , and therefore Bk ∩ lim supn∈S An = ∅, so lim supn∈S An ∈
/ I ∗.
(4) Clearly (P(X), {∅}) has the J-covering property iff |X| < non∗ (J). This fact
and the following reformulation of the J-covering property shows that covering
properties can be seen as “analytic star-uniformity”: (A, I) has the J-covering
property if and only if, for every (A, Borel([ω]ω ))-measurable function F : X →
S
[ω]ω (where X = A), there is an S ∈ J such that
x ∈ X : |F (x) ∩ S| < ω ∈ I.
(5) Notice that if (A, I) has the J-covering property, then (A[I], I) also has this
property where A[I] is the “I-completion of A”, that is
A[I] = B ⊆ X : ∃ A ∈ A A4B ∈ I .
For instance, Borel(R)[N] is the σ-algebra of Lebesgue measurable subsets of R,
and similarly Borel(R)[M] is the σ-algebra of sets with the Baire property.
(6) If (A, I) has the J-covering property then for all Y ∈ A\I the pair (A Y, I Y )
also has this property where of course, A Y = {Y ∩A : A ∈ A} is the restricted
σ-algebra.
(7) If both (A, I1 ) and (A, I2 ) have the J-covering property, then (A, I1 ∩I2 ) also has
this property. Moreover, it is easy to see that CPΓ (I1 ∩I2 ) = CPΓ (I1 )∩CPΓ (I2 ).
(8) How could we conclude a J1 -covering property from a J0 -covering property? It
is easy to see that if J0 ≤KB J1 and (A, I) has the J0 -covering property, then
(A, I) has the J1 -covering property as well. In other words, CP(I) is KB-upward
closed.
Now, let us discuss some negative results. We start from an easy example concerning
the measure case.
Example 2.3. We will show that N does not have the I1/n -covering property in a
strong sense. First consider interval (0, 1) and a fixed infinite-fold cover (An )n∈ω of
1
(0, 1) of the form An = (an , bn ), bn − an = n+1
. Then for each S ∈ I1/n we have
P
n∈S λ(An ) < ∞ where λ stands for Lebesgue measure on R. Hence by the BorelCantelli lemma, λ(lim supn∈S An ) = 0. Fix a homeomorphism h from (0, 1) onto R of
class C 1 . Then (h[An ])n∈ω is an open infinite-fold cover of R. Since h is absolutely
continuous, we have λ(lim supn∈S h[An ]) = 0, which gives the desired claim.
This example motivates the following question which will be discussed in Section 5
(see Example 5.1 and Theorem 5.2 below).
6
MAREK BALCERZAK, BARNABÁS FARKAS, AND SZYMON GLA̧B
Question 2.4. Assume X is a Polish space, I is a σ-ideal on X, and I does not have
the J-covering property. Does there exist an infinite-fold Borel cover (An )n∈ω of X such
that lim supn∈S An ∈ I for all S ∈ J?
We will show that a wide class of ideals cannot have any J-covering properties.
The following property was introduced in [1]: An ideal I on an uncountable Polish space X has property (M) if there is an uncountable Polish space Y and a Borelmeasurable f : X → Y such that f −1 [{y}] ∈ I + for each y ∈ Y . This property can be
seen as a strong violation of the countable chain condition. Recall that I is said to fulfil
this condition (or, that I is a ccc ideal) if every disjoint subfamily of Borel(X) \ I is
countable.
First of all, it is natural to ask about the existence of an ideal I on an uncountable
Polish space X without property (M) such that there is family of c pairwise disjoint
I-positive Borel sets. Surprisingly, this is still an open question in ZFC. D.H. Fremlin
constructed such an example under the assumption cov(M) = c (see [1, Prop. 1.5]).
Numerous classical ideals (without ccc) have property (M). Let us see some examples:
Example 2.5.
(1) The ideal of nowhere dense sets has property (M) (see [1, Prop. 1.3]).
(2) The σ-ideal Kσ , generated by compact sets on ω ω , has property (M). Indeed, if
F : ω ω → ω ω , F (x)(n) = x(2n) for each x ∈ ω ω and n ∈ ω, then F is continuous
ω
and F −1 [{y}] ∈ K+
σ for each y ∈ ω .
(3) The ideal of Hs -null subsets of R for s ∈ (0, 1) has property (M) where Hs is
the s-dimensional Hausdorff measure (see [11]).
(4) The ideal of so-called σ-porous subsets of R has property (M) (see [17, Prop.
2.1]).
(5) The so-called Mycielski ideal has property (M) (see [3, Lemma 0.1]).
(6) The ideal of so-called nowhere Ramsey subsets of [ω]ω has property (M) (see [1,
Prop. 1.4]).
(7) If J is a tall non prime ideal on ω, then let Jb be the ideal on [ω]ω generated by
the sets of the form
b = X ∈ [ω]ω : |A ∩ X| = ω for A ∈ J.
A
Clearly Jb is a σ-ideal iff J is a P-ideal. Let C0 ∪ C1 = ω be a partition into
J-positive sets, and let F : [ω]ω → 2ω , F (X)(n) = i iff the nth element of X
belongs to Ci . It is easy to see that F shows that Jb has property (M).
(8) It is easy to see that if I or J have property (M), then I ⊗ J also has this
property (see definition of the Fubini-product below).
The next simple result says that ideals with property (M) are not interesting in the
sense of covering properties.
COVERING PROPERTIES OF IDEALS
7
Proposition 2.6. If an ideal I on X has property (M ), then CP(I) = ∅. Moreover,
there is an infinite-fold Borel cover (Bn )n∈ω of X such that for each infinite and coinfinite set S ⊆ ω, lim supn∈S Bn ∈
/ I ∗.
Proof. Fix a Borel function f : X → ω ω which witnesses property (M ) of I. Consider
the following infinite-fold cover (An )n∈ω of ω ω by Fσ sets:
An = {x ∈ ω ω : x(n) 6= 0 or ∀∞ k x(k) = 0}.
It is easy to see that, if S ⊆ ω is infinite and co-infinite, then ω ω \ lim supn∈S An is
uncountable (even it is dense and does not belong to Kσ ). For each n, let Bn = f −1 [An ].
We claim that for each infinite and co-infinite set S ⊆ ω, lim supn∈S Bn ∈
/ I ∗ . Indeed,
h
i
lim sup Bn = f −1 lim sup An is co-uncountable,
n∈S
n∈S
hence the preimage of any x ∈
ωω
\ lim supn∈S An shows that lim supn∈S Bn ∈
/ I ∗.
As we will see, if we know that N has the J-covering property, then we can infer
the same for the Fubini product N ⊗ M, provided that J is a P-ideal on ω. For more
informations on the σ-ideals N ⊗ M and M ⊗ N on R2 , see [2].
We recall the definition of Fubini product of ideals I ⊆ P(X) and K ⊆ P(Y ). For
A ⊆ X × Y and x ∈ X let (A)x = {y ∈ Y : (x, y) ∈ A}, and let
I ⊗ K = A ⊆ X × Y : {x ∈ X : (A)x ∈ K} ∈ I ∗ .
It is easy to see that
CP(I ⊗ K) ⊆ CP(I) ∩ CP(K) = CP(I ∩ K)
for each I and K: Clearly, (Bn )n∈ω is an I-a.e. infinite-fold cover of X iff (Bn × Y )n∈ω
is an I ⊗ K-a.e. infinite-fold cover of X × Y . Similarly, (Cn )n∈ω is an K-a.e. infinite-fold
cover of Y iff (X × Cn )n∈ω is an I ⊗ K-a.e. infinite-fold cover of X × Y .
Theorem 2.7. CPP-ideals (N ⊗ M) = CPP-ideals (N).
Proof. To show the nontrivial inclusion “⊇”, assume that J is a P-ideal such that N has
the J-covering property. Let (An )n∈ω be an N ⊗ M-a.e. infinite-fold Borel cover of R2 .
By [2, Prop. 4], for each Borel set G in R2 there is a Borel set H with open sections
such that G4H ∈ N ⊗ M. So, we may assume that all sections (An )x for n ∈ ω and
x ∈ R are open. There exists a Borel set B ∈ N∗ such that lim supn∈ω (An )x is residual
for all x ∈ B. Fix a base {Uk : k ∈ ω} of open sets in R. For all n, k ∈ ω define
Dnk := {x ∈ R : (An )x ∩ Uk 6= ∅}.
Since (An )x ∩Uk 6= ∅ iff (An )x ∩Uk ∈
/ M, the sets Dnk are Borel (see [10, 22.22]). Observe
that for all k ∈ ω we have B ⊆ lim supn∈ω Dnk since for each x ∈ B there are infinitely
many An ’s such that ({x} × Uk ) ∩ An 6= ∅. Since N has the J-covering property, pick an
8
MAREK BALCERZAK, BARNABÁS FARKAS, AND SZYMON GLA̧B
Sk ∈ J such that lim supn∈Sk Dnk ∈ N∗ . Since J is a P-ideal, we can pick an S ∈ J such
that Sk ⊆∗ S for all k ∈ ω. Then
\
\
C :=
lim sup Dnk ⊇
lim sup Dnk ∈ N∗ .
k∈ω
n∈S
k∈ω
n∈Sk
Fix an x ∈ C. Then for all k ∈ ω and infinitely many indices n ∈ S, we have (An )x ∩Uk =
6
∅. It follows that lim supn∈S (An )x is a residual Gδ set for all x ∈ C. Hence (An )n∈S is
an N ⊗ M-a.e. infinite-fold cover of R2 .
Question 2.8. Is the analogous result true for M⊗N? Can one prove the same theorem
for all (Borel) ideals on ω?
Applying CP(I ⊗ K) ⊆ CP(I ∩ K), we obtain the following
Corollary 2.9. CPP-ideals (N) ⊆ CPP-ideals (M).
Finally in this section, we discuss the natural generalization of Elekes’ result about
random-indestructibility of Z.
Theorem 2.10. Let I be a σ-ideal on a Polish space X, and assume that PI is proper.
If I has the J-covering property, then J is PI -indestructible.
Proof. Assume on the contrary that Ẏ is a PI -name for an infinite subset of ω, i.e.
PI Ẏ ∈ [ω]ω and B PI ∀ A ∈ J |Ẏ ∩ A| < ω for some B ∈ PI . Then there are a
C ∈ PI , C ⊆ B, and a Borel function f : C → [ω]ω (coded in the ground model) such
that C PI f (ṙgen ) = Ẏ where ṙgen is a name for the generic real (see [18, Prop. 2.3.1]).
For each n ∈ ω let
Yn = f −1 [{S ∈ [ω]ω : n ∈ S}] ∈ Borel(X).
Then (Yn )n∈ω is an infinite-fold cover of C (by Borel sets) because x ∈ Yn iff n ∈ f (x)
and |f (x)| = ω. Using the J-covering property of (Borel(X) C, I C) we can choose
an A ∈ J such that (Yn )n∈A is an I-a.e. infinite-fold cover of C, that is |f (x) ∩ A| = ω
for I-a.e. x ∈ C, i.e. {x ∈ C : |f (x) ∩ A| < ω} ∈ I, so C PI |f (ṙgen ) ∩ A| = ω, and
consequently, C PI |Ẏ ∩ A| = ω, a contradiction.
By the following example, the covering property is, in general, truly stronger than
the forcing indestructibility.
Example 2.11. We show that ED and Fin ⊗ Fin are Cohen-indestructible but M does
not have the ED- or Fin ⊗ Fin-covering properties.
It is easy to see that non∗ (ED) = non∗ (Fin ⊗ Fin) = ω, so because of Observation
2.2(3), M does not have the corresponding covering properties.
Although ED ⊆ Fin ⊗ Fin, in particular Cohen-indestructibility of Fin ⊗ Fin follows
from the Cohen-indestructibility of ED, it is easy to see that a forcing notion P destroys
Fin ⊗ Fin iff P adds dominating reals.
COVERING PROPERTIES OF IDEALS
9
Cohen-indestructibility of ED: Let C = (2<ω , ⊇) be the Cohen forcing and assume
that Ẋ is a C-name for an infinite subset of ω × ω such that C ∃∞ n (Ẋ)n 6= ∅ (because
else Ẋ cannot destroy ED). Enumerate C = {pn : n ∈ ω}. By recursion on n ∈ ω we
will define A = {(mn , kn ) : n ∈ ω} ⊆ ω × ω (in the ground model). Assume (m` , k` )
is done for ` < n. Then we can choose a qn ≤ pn , an mn > mn−1 , and a kn such that
qn kn ∈ (Ẋ)mn . Clearly A ∈ ED.
We claim that C |A ∩ Ẋ| = ω. Assume on the contrary that p ∀ n ≥ N (A)n ∩
(Ẋ)n = ∅ for some p ∈ C and N ∈ ω. Then p = pn for some n and we can assume that
n ≥ N . Then mn ≥ n and qn kn ∈ (A)mn ∩ (Ẋ)mn , a contradiction.
3. Around the category case
In this section we will characterize CPBorel (M). We will need the following notion
and its characterizations: An ideal J on ω is called weak Q-ideal if for each partition
(Pn )n∈ω of ω into finite sets, there is an X ∈ J+ such that |X ∩ Pn | ≤ 1 for each n.
For example, the partition ([2n , 2n+1 ))n∈ω witnesses that Z is not a weak Q-ideal.
It is not hard to see that J is a weak Q-ideal iff EDfin KB J.
Note that EDfin ≤KB J holds for a quite big class of ideals, namely for tall analytic
P-ideals. This fact easily follows from Solecki’s representation theorem (see [15, Thm.
3.1.]) which says that analytic P-ideals are exactly the ideals of the form
n
o
Exh(ϕ) = A ⊆ ω : lim ϕ(A \ n) = 0
n→∞
for some lower semicontinuous submeasure ϕ on ω.
C. Laflamme in [12] proved another nice characterization of weak Q-ideals by using
infinite games. In general, he investigated the games of the form G(X, ω, Y) where
X, Y ⊆ P(ω). The rules are the following: At the kth stage Player I chooses an Xk ∈ X
and Player II responds with an nk ∈ Xk . Player II wins if {nk : k ∈ ω} ∈ Y. We will
need the characterization of existence of winning strategies in a game of this form.
Before the next theorem, we recall a notion: an ideal J on ω is ω-diagonalizable if
there is a countable family {Yn : n ∈ ω} of infinite and co-infinite subsets of ω such that
for each A ∈ J there is an n with A ⊆∗ Yn . Clearly, J is ω-diagonalizable iff non∗ (J) = ω,
in particular, in this case M does not have the J-covering property.
Theorem 3.1. ([12, Thm 2.2]) Let J be an ideal on ω. Then in the game G(Fin∗ , ω, J+ )
(1) Player I has a winning strategy iff J is not a weak Q-ideal;
(2) Player II has a winning strategy iff J is ω-diagonalizable.
Theorem 3.2. Assume J is a Borel ideal. Then M has the J-covering property iff J is
not a weak Q-ideal (i.e. EDfin ≤KB J). In other words,
CPBorel (M) = J : J is a Borel non weak Q-ideal .
10
MAREK BALCERZAK, BARNABÁS FARKAS, AND SZYMON GLA̧B
Proof. First we prove that M has the EDfin -covering property. It implies that CP(M)
contains all non weak Q-ideals. Let (A(n,m) )(n,m)∈∆ be an infinite-fold cover of R by
Borel sets. Without loss of generality, we can assume that all A(n,m) ’s are open and
nonempty.
Enumerate {Uk : k ∈ ω} a base of R. We will define by recursion a sequence
(nk , mk )k∈ω of elements of ∆. First, pick (n0 , m0 ) ∈ ∆ such that A(n0 ,m0 ) ∩ U0 6= ∅.
Assume (ni , mi ) are done for i < k. Then we can choose an (nk , mk ) ∈ ∆ such that
nk 6= ni for i < k and A(nk ,mk ) ∩ Uk 6= ∅. We obtain the desired set S = {(nk , mk ) : k ∈
S
ω} ∈ EDfin . For every k ∈ ω, the set i≥k A(ni ,mi ) is dense and open. Consequently,
lim sup(n,m)∈S A(n,m) is a dense Gδ set, hence it is residual.
Conversely, first we show that if J is Borel, then the game G(Fin∗ , ω, J+ ) is determined. We reformulate this game a bit. The Players are choosing elements from
the countable set Fin∗ × ω. The “set of rules” is the following tree on Fin∗ × ω:
((X0 , n0 ), (X1 , n1 ), . . . , (Xk−1 , nk−1 )) ∈ T iff n` ∈ X`−1 for each odd ` ∈ [1, k). Player I
wins the game ((Xk , nk ) : k ∈ ω) ∈ [T ] = {branches through T } if {nk : k is odd} ∈ J. It
is easy to see that the function from [T ] to P(ω), given by ((Xk , nk ) : k ∈ ω) 7→ {nk : k
is odd}, is continuous, and hence if I is Borel then the game is determined.
Applying Theorem 3.1, if EDfin KB J, then Player II has a winning strategy, so J is
ω-diagonalizable, and hence M does not have the J-covering property.
Notice that this theorem shows that M plays an important role in the following sense:
Corollary 3.3. If A \ I contains infinite antichains, then CPBorel (A, I) ⊆ CPBorel (M).
In particular, CPBorel (N) ⊆ CPBorel (M).
We also know that CP P-ideals (N) ⊆ CP P-ideals (M) (see Corollary 2.9), so it is natural
to ask the following:
Question 3.4. Does CP(N) ⊆ CP(M) hold?
4. KB-smallest elements in CP(I) and in CPBorel (I)
First we show that Z is not KB-minimal (in particular, it is not the KB-smallest ideal)
in CPBorel (N). For A ⊆ ω and n ∈ ω let
Sn (A) = max |A ∩ [k, k + n)| : k ∈ ω ,
and let
Sn (A)
=0 .
n→∞
n
Then Zu is an ideal called the uniform density zero ideal. It is easy to see that Zu is
an Fσδ non P-ideal (for more details, see [4]), and that EDfin ≤KB Zu ≤KB Z.
Zu =
Proposition 4.1. Z KB Zu .
A ⊆ ω : lim
COVERING PROPERTIES OF IDEALS
11
Proof. Let f : ω → ω be finite-to-one. We will inductively define a sequence (bn )n≥1 of
natural numbers such that
[
A=
f (bn + k) : n ∈ ω and k ∈ [0, n) ∈ Z.
Then f −1 [A] contains blocks of natural numbers of arbitrary length. Therefore f −1 [A] ∈
/
Zu so f cannot witness that Z ≤KB Zu .
Let b1 be such that f (b1 ) > 2 = 21 . Let b2 be such that f (b2 ), f (b2 + 1) > f (b1 ) + 22 .
Assume we have already defined b1 , . . . , bn . Let bn+1 be such that
min f (bn+1 + k) : k ∈ [0, n + 1) > max f (bn + k) : k ∈ [0, n) + 2n+1 .
S
Clearly, A =
f (bn + k) : n ∈ ω and k ∈ [0, n) has asymptotic density zero, that is,
A ∈ Z.
The proof of the following theorem is Elekes’ original proof, simply it works for Zu as
well.
Theorem 4.2. N has the Zu -covering property.
Proof. Let (An )n∈ω be an infinite-fold Borel cover of 2ω . Let N0 = 0. By continuity
of measure we can find A0 , A1 , . . . , AN1 −1 such that λ(2ω \ (A0 ∪ A1 ∪ · · · ∪ AN1 −1 )) <
1/2 (where λ is the Lebesgue measure on 2ω ). Note that if we remove finitely many
elements from an infinite-fold cover, then it is still an infinite-fold cover. Therefore we
can inductively define numbers 0 = N0 < N1 < N2 < . . . such that λ(2ω \ (ANk ∪
ANk +1 ∪ · · · ∪ ANk+1 −1 )) < 2−k−1 and k divides Nk − Nk−1 for any k > 0.
Since Nk − Nk−1 = dk k for some dk ∈ ω, we can define
Wik = {Nk−1 + i, Nk−1 + k + i, . . . , Nk−1 + (dk − 1)k + i}
for i ∈ [0, k), that is, the partition of [Nk−1 , Nk ) into k many arithmetical progressions
of length dk and difference k.
Q
Consider X = k≥1 [0, k) the product space of the uniformly distributed probability
spaces on k ≥ 1 with the discrete topology, and denote by µ the product probability
measure on this Polish space.
S
For each x̄ = (xk )k≥1 ∈ X let Z(x̄) = k≥1 Wxkk . Clearly Z(x̄) ∈ Zu for any x̄ ∈ X.
To end the proof we claim that (An )n∈Z(x̄) is a λ-a.e. infinite-fold cover of 2ω for µ-a.e.
x̄ ∈ X.
For a fixed f ∈ 2ω let
n
oo
[n
Ekf = x̄ ∈ X : f ∈
An : n ∈ Wxkk
∈ Borel(X).
Then µ(Ekf ) ≥ 1/k provided f ∈ ANk−1 ∪ · · · ∪ ANk −1 . But for λ-a.e. f ∈ 2ω there is
k0 (f ) such that f ∈ ANk−1 ∪ · · · ∪ ANk −1 for all k ≥ k0 (f ). Why? Simply because of the
12
MAREK BALCERZAK, BARNABÁS FARKAS, AND SZYMON GLA̧B
construction of (Nk )k∈ω we have


[ \
ANk−1 ∪ · · · ∪ ANk −1  = 1.
λ
k0 ≥1 k≥k0
P∞
f
f
ω
Therefore
k=1 µ(Ek ) = ∞ for λ-a.e. f ∈ 2 . Since (Ek )k∈ω are independent,
applying the second Borel-Cantelli Lemma we obtain that
f
ω
λ
f ∈ 2 : µ lim sup Ek = 1
= 1.
k→ω
And hence by Fubini’s theorem we are done.
Unfortunately, despite of the category case, the following question is still open.
Question 4.3. Does there exist a KB-smallest (or at least KB-minimal) element in
CPBorel (N)?
Recall that a sequence T = (Tα )α<κ in [ω]ω is a tower if κ is regular, T is ⊆∗ -descending
(i.e. Tβ ⊆∗ Tα if α < β < γ), and it has no pseudointersection. The tower number t is
the smallest cardinality of a tower, and c stands for the continuum.
Theorem 4.4. Assume t = c and |A| ≤ c. Then there is no KB-smallest element in
CP(A, I).
Proof. If CP(A, I) = ∅ then we are done. If (A, I) has the J0 -covering property then we
will construct a J such that J0 KB J but (A, I) has the J-covering property.
Enumerate (fα )α<c all finite-to-one functions from ω to ω, and enumerate ((Aαn )n∈ω :
S
α < c) the infinite-fold covers of X = A by sets from A. By recursion on c we will
define a ⊆∗ -increasing sequence (Sξ )α<c of infinite and co-infinite subsets of ω and the
ideal J generated by this sequence will be as required.
Assume (Sξ )ξ<α is done for some α < c. Because of our assumption on t, we can
choose an infinite and co-infinite Sα0 such that Sξ ⊆∗ Sα0 for each ξ < α. The set
fα [ω \ Sα0 ] contains an infinite element E of J0 . We want to guarantee that fα−1 [E] ∈
/J
−1
0
ω
because then fα can not witness J0 ≤KB J. Let H = fα [E] \ Sα ∈ [ω] .
Consider the αth cover (Aαn )n∈ω . If (An )n∈ω\H is an I-a.e. infinite-fold cover of X,
then let Sα = Sα0 ∪ (ω \ H).
If not, then
C = x ∈ X : {n ∈ ω \ H : x ∈ Aαn } is finite ∈
/ I.
Applying our assumption for (A[I] C, I C) and (Aαn ∩ C)n∈H (with a copy of J0 on
H) we can choose an infinite H 0 ⊆ H such that H \ H 0 is also infinite and (Aαn ∩ C)n∈H 0
is an I C-a.e. infinite-fold cover of C.
Finally, let Sα = Sα0 ∪ (ω \ H) ∪ H 0 . It is easy to see from the construction that J is
as required.
COVERING PROPERTIES OF IDEALS
13
Corollary 4.5. If t = c then there are ideals J0 ∈ CP(N) and J1 ∈ CP(M) such that
Zu KB J0 and EDfin KB J1 .
Without t = c we can use a simple forcing construction.
Theorem 4.6. After adding ω1 Cohen-reals, there is an ideal J such that EDfin KB J
(in particular, Zu KB J) but N and M have the J-covering property (i.e. J ∈ CP(N) ∩
CP(M) = CP(N ∩ M)).
Proof. Let (cα )α<ω1 be the sequence of generic Cohen-reals in 2ω , Cα = c−1
α [{1}] ⊆ ω,
and let J be the ideal generated by these sets. J is a proper ideal because it is well-known
that {Cα : α < ω1 } is an independent system of subsets of ω.
To show that N has the J-covering property in the extension, it is enough to see that if
(An )n∈ω is an infinite-fold cover of 2ω by Borel sets in a ground model V , then (An )n∈C
is an N-a.e. infinite-fold cover of 2ω in V [C] where C ⊆ ω is a Cohen-real over V .
S
Clearly, it is enough to prove that V [C] |= λ n∈C\k An = 1 for each k ∈ ω (because
then V [C] |= λ lim supn∈C An = 1). Let p ∈ C = (2<ω , ⊇), k ∈ ω, and ε < 1. We
S
can assume that |p| ≥ k. Then there is an m ≥ |p| such that λ n∈m\|p| An > ε so if
S
q : m → 2, q |p| = p, and q (m \ |p|) ≡ 1, then q ≤ p and q λ n∈Ċ\k An > ε.
To show that M has the J-covering property in the extension, it is enough to prove
that if (An )n∈ω is an infinite-fold cover of 2ω by open sets in V , then (An )n∈C is an MS
a.e. infinite-fold cover of 2ω in V [C]. By a simple density argument V [C] |=“ n∈C\k An
is dense open” for each k ∈ ω, so V [C] |=“lim supn∈C An is residual.”
To show that EDfin KB J, it is enough to see that if f ∈ ∆ω ∩ V [(cξ )ξ<α ] is a
finite-to-one function for some α < ω1 , then there is an A ∈ EDfin ∩ V [(cξ )ξ≤α ] such
that f −1 [A] cannot be covered by finitely many of Cξ ’s (ξ < ω1 ). Simply let A be a
Q
Cohen function in n∈ω (n + 1), i.e. the graph of a Cohen-function in ∆, for example
A = (n, k) ∈ ∆ : c0α (n) ≡ k mod (n + 1) is suitable. Using the presentation of this
iteration by finite partial functions from ω1 × ω to 2, we are done by a simple density
argument.
Question 4.7. Is it provable in ZFC that there are no KB-smallest elements of CP(N)
and CP(M)? Or at least, is it provable that Zu and EDfin are not the KB-smallest
members of these families?
The next result shows that, to give positive answer to the second question, it would be
enough to find a non meager weak Q-ideal. However, at this moment we are unable to
construct such an ideal. Discussing the existence of a non meager weak Q-ideal could be
interesting on its on right because the so-called Q-ideals satisfy these properties (these
are the most natural consequences of property Q) but it is consistent that there are no
Q-ideals (see [14]).
Proposition 4.8. CP(M) ∩ CP(N) contains all non meager ideals.
14
MAREK BALCERZAK, BARNABÁS FARKAS, AND SZYMON GLA̧B
Proof. Let J be non meager, that is (by Theorem 1.2), for each partition (Pn )n∈ω of ω
into finite sets, there is an S ∈ J which contains infinite many elements of the partition.
We claim that J ∈ CP(M). Let (An )n∈ω be an M-a.e. infinite-fold Borel cover of R.
We can assume that An ’s are open. Fix a base {Uk : k ∈ ω} of R. By recursion on n we
can define a partition (Pn )n∈ω of ω into finite sets such that for each k ≤ n
[
Uk ∩
Ai : i ∈ Pn 6= ∅.
We can do it because for each N the sequence (An )n≥N is still an M-a.e. infinite-fold
cover of R. By our assumption on J, there is an S ∈ J such that Pn ⊆ S for infinitely
S
many n’s. It yields that
An : n ∈ S \ N is a dense open set for each N , hence
lim supn∈S An is residual.
We claim that J ∈ CP(N). Let (An )n∈ω be an N-a.e. infinite-fold Borel cover of R.
By recursion on n we can define a partition (Pn )n∈ω of ω such that for each n
[
> 2n − 2−n .
λ [−n, n] ∩
Ai : i ∈ P n
We can do it because for each N the sequence (An )n≥N is still an N-a.e. infinite-fold
cover. Applying our assumption on J, there is an S ∈ J such that Pn ⊆ S for infinitely
S
many n’s. It yields that
An : n ∈ S\N is co-null for each N and hence lim supn∈S An
is also co-null.
5. When the J-covering property “strongly” fails
In this section we give a positive answer to Question 2.4 in a special case. First of
all, we present a counterexample:
Example 5.1. Consider X = (−1, 1) and let an ideal I on X consist of sets A ⊆ X such
that A ∩ (−1, 0] is meager and A ∩ (0, 1) is of Lebesgue measure zero. Using Example
2.3 and Theorem 3.2 observe that I yields the negative answer to Question 2.4 with
J = I1/n . However, this question remains interesting if we restrict it to translation
invariant ideals on R. In this case, we describe a class of ideas which yields a positive
answer to Question 2.4 provided J is a P-ideal.
Let Q stand for the set of rational numbers. For A, B ⊆ R and x ∈ R we write
A + x = {a + x : a ∈ A} and A + B = {a + b : A ∈ A and b ∈ B}.
Theorem 5.2. Assume that I is a translation invariant ccc σ-ideal on R fulfilling the
condition
(1)
Q + A ∈ I ∗ for each A ∈ Borel(R) \ I.
Fix a P-ideal J on ω. If I does not have the J-covering property, then there exists an
infinite-fold Borel cover (A0n )n∈ω of R with lim supn∈S A0n ∈ I for all S ∈ J.
COVERING PROPERTIES OF IDEALS
15
Proof. Fix an infinite-fold Borel cover (An )n∈ω of R such that lim supn∈S An ∈
/ I ∗ for all
S ∈ J. We will show that there is a Borel set B ⊆ R with B ∈
/ I and (lim supn∈S An )∩B ∈
I for all S ∈ J. Suppose it is not the case. So, in particular (when B = R), we find
S0 ∈ J with X0 := lim supn∈S0 An ∈
/ I. Then by transfinite recursion we define sequences
S
(Sα )α<γ and (Xα )α<γ with Sα ∈ J and Xα := (lim supn∈Sα An ) \ β<α Xβ ∈
/ I (when
S
B = R \ β<α Xβ ∈
/ I). Since I is ccc, this construction stops at a stage γ < ω1 with
S
S
∗
α<γ lim supn∈Sα An =
α<γ Xα ∈ I . Since I is a P-ideal, there is S ∈ J which almost
contains each Sα for α < γ. Then lim supn∈S An ∈ I ∗ which contradics our supposition.
So, fix a Borel set B ∈
/ I such that (lim supn∈S An ) ∩ B ∈ I for all S ∈ J. Let
S
Q = {qk : k ∈ ω}. Define B0 := B and Bk := (qk + B) \ i<k Bi for k ∈ ω. Then put
S
A0n := k∈ω ((qk + An ) ∩ Bk ) for n ∈ ω. Since (An )n∈ω is an infinite-fold cover of R,
we have lim supn∈ω ((qk + An ) ∩ Bk ) = Bk for all k ∈ ω. Note that (A0n )n∈ω is an I-a.e.
infinite-fold cover of R since
[
lim sup A0n ⊇
lim sup((qk + An ) ∩ Bk ) = Q + B
n∈ω
k∈ω
n∈ω
and Q + B ∈ I ∗ by (1). Finally, let S ∈ J. Since I is translation invariant and
(lim supn∈S An ) ∩ B ∈ I, we have lim supn∈S ((qk + An ) ∩ Bk ) ∈ I for all k ∈ ω. Since I
is a σ-ideal and Bk ’s are pairwise disjoint, it follows that
[
lim sup A0n =
lim sup((qk + An ) ∩ Bk ) ∈ I.
n∈S
k∈ω
n∈S
Of course, we can modify (A0n ) to be an infinite-fold cover of R.
Theorem 5.2 can be generalized to any Polish group G with Q replaced by a countable
dense subset of G. Condition (1) is related to the Steinhaus property, for details see [5].
Note that M, N, M ⊗ N and N ⊗ M satisfy (1) with Q replaced by any dense subset of
R (resp. R2 ).
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Institute of Mathematics, Technical University of Lódź, ul. Wólczańska 215, 93-005
Lódź, Poland
E-mail address: marek.balcerzak@p.lodz.pl
Institute of Mathematics, Budapest University of Technology and Economics, Hungary
E-mail address: barnabasfarkas@gmail.com
Institute of Mathematics, Technical University of Lódź, ul. Wólczańska 215, 93-005
Lódź, Poland
E-mail address: szymon.glab@p.lodz.pl